Remarks on the complete synchronization of Kuramoto oscillators

We present an improved exponential frequency synchronization estimate for globally coupled Kuramoto oscillators. For a sufficiently large coupling, it is numerically observed that Kuramoto oscillators exhibit relaxation toward the phase-locked state, independent of the initial configuration. This phenomenon has never been confirmed analytically in full generality. To date, the analytical treatment of complete frequency synchronization is restricted to initial configurations that are geometrically confined to the half-unit circle. We extend this previous work (Choi et al 2012 Physica D 241 735-54 and Chopra and Spong 2009 IEEE Trans. Autom. Control 54 353-7) to a class of initial configurations lying on an arc of length greater than pi by exploiting the dynamics of the Kuramoto order parameter in a finite-dimensional setting.